CONVERGENCE ANALYSIS FOR DOUBLE PHASE OBSTACLE PROBLEMS WITH MULTIVALUED CONVECTION TERM

. In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by S the solution set of the obstacle problem and by S n the solution sets of approximating problems, we prove the following convergence relation


Introduction
Recently, based on a surjectivity result for pseudomonotone operators obtained by Le [25], the authors [44] have studied the nonemptyness, boundedness and closedness of the set of weak solutions to the following double phase problem with a multivalued convection term and obstacle effect − div |∇u| p−2 ∇u + µ(x)|∇u| q−2 ∇u ∈ f (x, u, ∇u) in Ω, in Ω, where Ω ⊆ R N is a bounded domain with Lipschitz boundary ∂Ω, 1 < p < q < N , µ : Ω → [0, ∞) is Lipschitz continuous, f : Ω × R × R N → 2 R is a multivalued function depending on the gradient of the solution and Φ : Ω → R + is a given function, see Section 3 for the precise assumptions.
As the obstacle effect leads to various difficulties in obtaining the exact and numerical solutions, it is reasonable to consider some appropriate approximating methods to overcome/avoid the obstacle effect. In the present paper, we are going to propose a family of approximating problems corresponding to (1.1) and deliver an important convergence theorem which indicates that the solution set of the obstacle problem can be approximated by the solutions of perturbation problems. More precisely, let {ρ n } be a sequence of positive numbers such that ρ n → 0 as n → ∞ and for each n ∈ N, we consider the following problem − div |∇u| p−2 ∇u + µ(x)|∇u| q−2 ∇u in Ω, u = 0 on ∂Ω. (1.2) Denoting by S and S n the sets of solutions to problems (1.1) and (1.2), respectively, we shall establish the relations between the sets S, w-lim sup n→∞ S n (being the weak Kuratowski upper limit of S n ) and s-lim sup n→∞ S n (being the strong Kuratowski upper limit of S n ), see Definition 2.2.
The introduction of so-called double phase operators goes back to Zhikov [46] who described models of strongly anisotropic materials by studying the functional u → (|∇u| p + µ(x)|∇u| q ) dx. (1. 3) The integral functional (1.3) is characterized by the fact that the energy density changes its ellipticity and growth properties according to the point in the domain. More precisely, its behavior depends on the values of the weight function µ(·). Indeed, on the set {x ∈ Ω : µ(x) = 0} it will be controlled by the gradient of order p and in the case {x ∈ Ω : µ(x) = 0} it is the gradient of order q. This is the reason why it is called double phase. Functionals of the expression (1.3) have been studied more intensively in the last five years. Concerning regularity results, we refer, for example, to the works of Baroni-Colombo-Mingione [4,5,6], , Cupini-Marcellini-Mascolo [15], Colombo-Mingione [13], [14], Marcellini [28,29] and the references therein.
Double phase differential operators and corresponding energy functionals appear in several physical applications. For example, in the elasticity theory, the modulating coefficient µ(·) dictates the geometry of composites made of two different materials with distinct power hardening exponents q and p, see Zhikov [47]. We also refer to other applications which can be found in the works of Bahrouni-Rȃdulescu-Repovš [1] on transonic flows, Benci-D'Avenia-Fortunato-Pisani [8] on quantum physics and Cherfils-Il yasov [9] on reaction diffusion systems.
The paper is organized as follows. In Section 2 we recall the definition of the Musielak-Orlicz spaces L H (Ω) and its corresponding Sobolev spaces W 1,H (Ω) and we recall the definition of the Kuratowski lower and upper limit, respectively. In Section 3 we present the full assumptions on the data of problem (1.2), give the definition of weak solutions for (1.1) as well as (1.2) and state and prove our main result, see Theorem 3.4.

Preliminaries
Let Ω be a bounded domain in R N and let 1 ≤ r ≤ ∞. In what follows, we denote by L r (Ω) := L r (Ω; R) and L r (Ω; R N ) the usual Lebesgue spaces endowed with the norm · r . Moreover, W 1,r (Ω) and W 1,r 0 (Ω) stand for the Sobolev spaces endowed with the norms · 1,r and · 1,r,0 , respectively. For any 1 < r < ∞ we denote by r the conjugate of r, that is, 1 r + 1 r = 1. For the weight function µ and powers p, q we will assume that: H(µ): µ : Ω → R + := [0, ∞) is Lipschitz continuous and 1 < p < q < N are chosen such that We consider the function H : Ω × R + → R + defined by We know that L H (Ω) is uniformly convex and so a reflexive Banach space. In addition, we introduce the seminormed function space which is equipped with the seminorm · q,µ given by It is known that the embeddings for all u ∈ L H (Ω). By W 1,H (Ω) we denote the corresponding Sobolev space which is defined by equipped with the norm .
Furthermore, we have the following compact embedding for each 1 < r < p * , where p * is the critical exponent to p given by Let us now consider the eigenvalue problem for the negative r-Laplacian with homogeneous Dirichlet boundary condition and 1 < r < ∞ which is defined by in Ω, From Lê [26] we know that the set σ r being the set of all eigenvalues of − ∆ r , W 1,r 0 (Ω) has a smallest element λ 1,r which is positive, isolated, simple and it can be variationally characterized through (Ω) * be the operator defined by (Ω), where ·, · H is the duality pairing between W 1,H 0 (Ω) and its dual space (Ω) * can be summarized as follows, see Liu-Dai [27].
Proposition 2.1. The operator A defined by (2.6) is bounded, continuous, monotone (hence maximal monotone) and of type (S + ).
Throughout the paper the symbols " " and "→" stand for the weak and the strong convergence, respectively. Let (V, · V ) be a Banach space with its dual V * and denote by ·, · the duality pairing between V * and V . We end this section by recalling the following definition, see, for example, Papageorgiou-Winkert [38, Definition 6.7.4].
Definition 2.2. Let (X, τ ) be a Hausdorff topological space and let {A n } ⊂ 2 X be a sequence of sets. We define the τ -Kuratowski lower limit of the sets A n by x n , x n ∈ A n for all n ≥ 1 , and the τ -Kuratowski upper limit of the sets A n τ -lim sup then A is called τ -Kuratowski limit of the sets A n .

Main results
We assume the following hypotheses on the data of problem (1.2).
and convex values such that (i) the multivalued mapping x → f (x, s, ξ) has a measurable selection for all (s, ξ) ∈ R × R N ; (ii) the multivalued mapping (s, ξ) → f (x, s, ξ) is upper semicontinuous for almost all (a. a.) x ∈ Ω; (iii) there exists α ∈ L q 1 q 1 −1 (Ω) and a 1 , a 2 ≥ 0 such that for all η ∈ f (x, s, ξ), for a. a. x ∈ Ω, all s ∈ R and all ξ ∈ R N , where 1 < q 1 < p * with the critical exponent p * given in (2.4); (iv) there exist w ∈ L 1 + (Ω) and b 1 , b 2 ≥ 0 such that for all η ∈ f (x, s, ξ), for a. a. x ∈ Ω, all s ∈ R and all ξ ∈ R N , where λ 1,p is the first eigenvalue of the Dirichlet eigenvalue problem for the p-Laplacian, see (2.5).
(3.1)  (a) We say that u ∈ K is a weak solution of problem It is straightforward, to prove the following lemma.

2)
is bounded, demicontinuous and monotone, where ·, ·, q1 denotes the duality pairing between L q1 (Ω) and its dual space L q 1 (Ω). Now, we can state the main result of this paper. (Ω) * . Since 1 < q 1 < p * the embedding operator i is compact and so i * as well. From hypotheses H(f )(i) and (iii), we see that the Nemytskij operator N f : W 1,H 0 (Ω) ⊂ L q1 (Ω) → 2 L q 1 (Ω) associated to the multivalued mapping f given by (Ω) is well-defined (see the proof of Proposition 3 in Papageorgiou-Vetro-Vetro [36]). The convexity and closedness of the values of f ensure that N f has closed and convex values as well. Moreover, by hypothesis H(f )(iv) we have (Ω) and η ∈ N f (u) such that (Ω) * and B : L q1 (Ω) → L q 1 (Ω) are given by (2.6) and (3.2), respectively.
Then, using the same arguments as in the proof of Zeng-Gasiński-Winkert-Bai [44, Theorem 3.3], we can conclude that for each n ∈ N, the set S n of solutions to problem (1.2) is nonempty, bounded and closed.
(ii) First, we prove that the set w-lim sup n→∞ S n is nonempty. Indeed, we have the following (Ω) with u n ∈ S n for each n ∈ N such that u n 1,H,0 → ∞ as n → ∞.
Hence, for each n ∈ N, we are able to find η n ∈ N f (u n ) such that Ω |∇u n | p−2 ∇u n + µ(x)|∇u n | q−2 ∇u n · ∇v dx + 1 (Ω). Inserting v = u n into the inequality above, we get Ω |∇u n | p−2 ∇u n + µ(x)|∇u n | q−2 ∇u n · ∇u n dx − By the nonnegativity of Φ and the monotonicity of the function s → s + , we have However, by hypothesis H(f )(iv), we have where the last inequality is obtained by (2.2). Since 1 < p < q < N and b 1 + b 2 λ −1 1,p < 1, we can take R 0 > 0 large enough such that for all R ≥ R 0 it holds Therefore, we are able to find N 0 > 0 large enough such that u n 1,H,0 > R 0 for all n ≥ N 0 and (Ω) with u n ∈ S n for each n ∈ N be an arbitrary sequence. Claim 1 indicates that {u n } is bounded in W 1,H 0 (Ω). Then, we may assume that along a relabeled subsequence we have u n u as n → ∞ (3.6) for some u ∈ W 1,H 0 (Ω). This guarantees that the set w-lim sup n→∞ S n is nonempty.
Next, we are going to demonstrate that w-lim sup n→∞ S n is a subset of S. Let u ∈ w-lim sup n→∞ S n be arbitrary. Without loss of generality, we may suppose that there exists a subsequence {u n } ⊂ W 1,H 0 (Ω) with u n ∈ S n for all n ∈ N, satisfying (3.6). Our goal is to prove that u ∈ S. Claim 2. u(x) ≤ Φ(x) for a.a. x ∈ Ω. For every n ∈ N, we have It follows from Hölder's inequality and (3.3) that Putting (3.8) into (3.7), employing the boundedness of A (see Proposition 2.1), the convergence (3.6), and the embedding (2.3), we have (Ω). Passing to the limit in the inequality above, using convergence (3.6), the compact embedding (2.3), and the Lebesgue Dominated Convergence Theorem, we conclude that Claim 3. u ∈ S. For each n ∈ N, we have (Ω). The latter combined with the monotonicity of s → s + gives for all v ∈ W 1,H 0 (Ω). Hence, for all v ∈ K, where K is defined in (3.1).
Claim 2 indicates that u ∈ K, so, we put v = u in (3.9) to obtain It follows from the proof of Theorem 3.3 in Zeng-Gasiński-Winkert-Bai [44] that the multivalued mapping This means that for each v ∈ K, there is an element η(v) ∈ N f (u) satisfying For each v ∈ K, passing to the lower limit as n → ∞ in inequality (3.9), we are able to find an element η(v) ∈ N f (u) such that We shall prove that u ∈ K is a weak solution to problem (1.1), namely, there exists an element η * ∈ N f (u), which is independent of v, such that for all v ∈ K. Arguing by contradiction, suppose that for each For any v ∈ K, let us consider the set R v ⊂ N f (u) defined by We now assert that for each v ∈ K, the set R v is weakly open. Let {η n } ⊂ R c v be such that η n η for some η ∈ L q 1 (Ω) as n → ∞, where R c v denotes the complement of R v . Hence, for all n ∈ N. Passing to the limit in the inequality above, we obtain that η ∈ R c v . Therefore, for every v ∈ K, the set R v is weakly open in L q 1 (Ω). Besides, we observe that {R v } v∈K is an open covering of N f (u). The latter coupled with the facts that L q1 (Ω) is reflexive and N f (u) is weakly compact and convex in L q 1 (Ω), ensures that {R v } v∈K has a finite sub-covering of N f (u) , let us say {R v1 , R v2 , . . . , R vn } for some points {v 1 , v 2 , . . . , v n } ⊆ K. Let κ 1 , κ 2 , . . . , κ n be a partition of unity for N f (u), where for each i = 1, 2, . . . , n, κ i : N f (u) → [0, 1] is a weakly continuous function such that Obviously, the function M is also weakly continuous due to the weak continuity of κ i for i = 1, 2, . . . , n. For any η ∈ N f (u), we have for all η ∈ N f (u), where the last inequality is obtained by the use of Lemma 7.3(ii) of Granas-Dugundji [23]. Let us define two multivalued functions Λ : K → 2 N f (u) and Ψ : for all v ∈ K, and Ψ(η) := Λ(M(η)) for all η ∈ N f (u).
Then, Ψ has nonempty, weakly compact and convex values (by (3.10) and because N f (u) is bounded closed and convex in L q 1 (Ω)) and Λ is upper semicontinuous from the normal topology of K to weak topology of L q 1 (Ω). From Migórski-Ochal-Sofonea [31, Proposition 3.8], it is enough to verify that for each weakly closed set D in L q 1 (Ω), the set be a sequence such that v n → v as n → ∞. Then, for each n ∈ N, we are able to find η n ∈ N f (u) satisfying From the weak compactness of N f (u), without any loss of generality, we may suppose that η n η in L q 1 (Ω), as n → ∞, for some η ∈ N f (u). Passing to the upper limit as n → ∞ for (3.13), we have that is, η ∈ Λ(v). But, the weak closedness of D implies that η ∈ D. Therefore, η ∈ Λ(v) ∩ D and so v ∈ Λ − (D). Applying Migórski-Ochal-Sofonea [31, Proposition 3.8] derives that Λ is strongly-weakly upper semicontinuous. On the other hand, the continuity of M and Theorem 1.2.8 of Kamenskii-Obukhovskii-Zecca [24] imply that Ψ is also strongly-weakly upper semicontinuous.
We are now in a position to employ Tychonov fixed point principle, (see, for example, Granas-Dugundji [23, Theorem 8.6]) for function Ψ, to conclude that there exists η ∈ N f (u) such that This leads to a contraction with (3.12). Consequently, we infer that u ∈ K solves problem (1.1) as well, that means, there exists η ∈ N f (u), which is independent of v, such that (3.11) holds.
Consequently, we conclude that ∅ = w-lim sup n→∞ S n ⊂ S. Since s-lim sup n→∞ S n ⊂ w-lim sup n→∞ S n , it is enough to verify the condition w-lim sup n→∞ S n ⊂ s-lim sup n→∞ S n . Let u ∈ w-lim sup n→∞ S n be arbitrary. Without any loss of generality, there exists a sequence, still denoted by {u n } with u n ∈ S n such that u n u as n → ∞. We claim that u n → u as n → ∞. For each n ∈ N, it holds Au n , u n − v H = − Ω (u n (x) − Φ(x)) + (u n (x) − v(x)) dx + Ω η n (x)(u n (x) − v(x)) dx for some η n ∈ N f (u n ) and for all v ∈ W 1,H 0 (Ω). Inserting v = u into the above inequality and passing to the upper limit as n → ∞ for the resulting inequality, we can use the compact embedding (2.3) to get lim sup n→∞ Au n , u n − u H ≤ 0.
The latter combined with the convergence u n u as n → ∞ and the fact that A is of type (S + ) (see Proposition 2.1) implies that u n → u as n → ∞. This means that u ∈ s-lim sup n→∞ S n . Therefore s-lim sup n→∞ S n = w-lim sup n→∞ S n .
(iii) Let u ∈ s-lim sup n→∞ S n be arbitrary. Since S n is nonempty, bounded and closed, so, the set T (S n , u) is nonempty. Let { u n } be any sequence such that u n ∈ T (S n , u) for each n ∈ N.
It follows from Claim 1 that the sequence { u n } is bounded. So, passing to a subsequence, we may assume, that u n u as n → ∞ for some u ∈ W 1,H 0 (Ω). Thus, using the same argument as the proof of Claim 2, we get that u ∈ K. Then, for each n ∈ N, we have (Ω). Proceeding in the same way as in the proof of Claim 3, we conclude that u is a solution to problem (1.1) as well. Consequently, the desired conclusion is proved.